The impact of policing and homelessness on violence experienced by women who sell sex in London: a modelling study

Street-based sex workers experience considerable homelessness, drug use and police enforcement, making them vulnerable to violence from clients and other perpetrators. We used a deterministic compartmental model of street-based sex workers in London to estimate whether displacement by police and unstable housing/homelessness increases client violence. The model was parameterized and calibrated using data from a cohort study of sex workers, to the baseline percentage homeless (64%), experiencing recent client violence (72%), or recent displacement (78%), and the odds ratios of experiencing violence if homeless (1.97, 95% confidence interval 0.88–4.43) or displaced (4.79, 1.99–12.11), or of experiencing displacement if homeless (3.60, 1.59–8.17). Ending homelessness and police displacement reduces violence by 67% (95% credible interval 53–81%). The effects are non-linear; halving the rate of policing or becoming homeless reduces violence by 5.7% (3.5–10.3%) or 6.7% (3.7–10.2%), respectively. Modelled interventions have small impact with violence reducing by: 5.1% (2.1–11.4%) if the rate of becoming housed increases from 1.4 to 3.2 per person-year (Housing First initiative); 3.9% (2.4–6.9%) if the rate of policing reduces by 39% (level if recent increases had not occurred); and 10.2% (5.9–19.6%) in combination. Violence reduces by 26.5% (22.6–28.2%) if half of housed sex workers transition to indoor sex work. If homelessness decreased and policing increased as occurred during the COVID-19 pandemic in 2020, the impact on violence is negligible, decreasing by 0.7% (8.7% decrease-4.1% increase). Increasing housing and reducing policing among street-based sex workers could substantially reduce violence, but large changes are needed.

), and a Monte Carlo Markov chain (MCMC) was used to search this parameter space, with each step in the MCMC (set of parameter values) used to run the model and output summary statistics ( ' ), which were compared to a set of target summary statistics.The ABC-MCMC analysis was done in R (v3.6.1) using the Wegmann 1 method within the package EasyABC (v1.5) 2 .First, the MCMC was run for 10,000 steps, and this was used to calculate a threshold for closeness of fit between output and target summary statistics above which 1% of parameter sets would be accepted.This threshold value was then used to accept or reject parameter sets drawn in next phase of the MCMC, until 1,000 parameter sets were accepted, with every 5 steps in the chain compared to the target summary statistics and accepted or rejected.In addition, a Box-Cox transformation was applied to each of the summary statistics and partial-least squares regression was used to define components in order to reduce correlation between the summary statistics in terms of their predictive value for model parameters and use the most informative target values.For each model run, the first 15 years of simulation were discarded prior to changing parameters to ensure models had reached equilibrium before introducing changes.

Prior assumptions for parameters
Prior distributions for the proportion that transition between groups were based on an uninformative Beta(1,1) prior (equivalent to a uniform distribution between 0 and 1).The Beta distribution was updated with observed data from the cohort study by adding the number that transitioned to the first shape parameter and the number that didn't transition to the second shape parameter.This results in a prior distribution between 0 and 1 that is centred on the observed proportion that transitioned and the standard deviation of the distribution decreases as the sample size of the observations increase.The parameters representing the proportion transitioning are then converted to annual transition rates within the model by −ln (1 − )/0.5.
This method of prior estimation was used for three parameters based on the cohort data: the rate of transitioning to recent violence category if not recently displaced or homeless (l), the rate of transitioning to recent policing category if not homeless (h), and the rate of leaving homelessness (a).For l and h, we looked only at individuals who were in the relevant risk category (not recently displaced nor homeless, or not homeless, respectively) at both baseline and follow.Of these, we calculated the proportion who reported the relevant outcome (recent violence or recent police displacement), at follow up.This represents the proportion who experienced the outcome while remaining consistently in the same (lower) risk category.For the rate of leaving homelessness, we calculated the number of participants who transitioned from homelessness to housing between the baseline and follow up surveys.

Indoor sex work model parameterisation
For each of the 1000 parameter sets from the baseline model, we estimated two additional parameters.First, as the possibility of transitioning to indoor sex work is uncertain, we set a target value of 5%, 50%, or 75% for the proportion of the total population who transition to indoor sex work (Z0 + Z1 at equilibrium).We then fixed  at 0 and tested a sequence of values for  ranging from 0.001 to 0.1 in increments of 0.001, and selected the value of  which minimized the distance from 5%, 50%, or 75% for the proportion of modelled individuals in Z compartments.The range was of tested parameters was expanded until the target values could be reached.By incrementing parameters rather than fitting precisely to the target values, we allow for an uncertainty range around the target proportions.
We then used a similar method to conduct a second step fitting procedure for .In each parameter set for the 5%, 50%, and 75% scenarios, we set a target value for the proportion of indoor sex workers experiencing recent violence Z1 /(Z0 + Z1) to be 36%, as observed in the East London Project.The sequence of values tested for  ranged from 0.1 to 0.6 in increments of 0.01.
The resulting parameter distributions are: for the 5% target,  ranges from 0.002-0.005and  is 0.42-0.43resulting in 4.0-6.6%transitioning to indoor sex work, with 35.6-36.4% of indoor sex workers experiencing recent violence; for the 50% target,  ranges from 0.020-0.063and  is always 0.43 resulting in 49.

Sensitivity analysis
An alternative version of the model was tested by changing the fixed rate of leaving recent violence (g) and rate of leaving recent displacement (d) to 1 or 2.5 (compared to baseline with both values fixed at 2).The model was re-fit allowing all parameters to adjust alongside the new values for the fixed parameters and the PAF was calculated as in the main analysis.We tested using larger values, however setting the parameters to 3 or greater resulted in failure of the ABC fitting process.

Variance explained
To assess what drives differences in each PAF value, we conducted linear regression analyses with each of the PAF calculations (remove homelessness, remove displacement, remove both) as dependent variables, and the six target summary statistics as independent variables, included in the model in the order they are listed in the main text methods.We then used the anova() function in R to calculate the variance explained by each predictor as the incremental sum of squares.

Sensitivity analysis
Alternative model fits setting the rates of leaving recent violence and recent displacement to 1 resulted in only 238 out of 1000 parameter sets fitting within the uncertainty bounds of all target summary statistics.Under this scenario, the proportion reduction in violence when homelessness is eliminated increased to 40.5% (95% CrI 18.
4%) in the base case.For removing both homelessness and displacement, the reduction in violence again was similar, at 72.9% (95% CrI 53.9-86.3%)compared to 67.3% (95% CrI 53.1-81.1%) in the base case.Increasing the rates of leaving recent violence and recent displacement to 2.5 resulted in 543/1000 parameter sets fitting within the uncertainty bounds of all target summary statistics.The reduction in violence when homelessness or displacement are removed both decreases slightly compared to the base case, with 25.3% (95% CrI 7.7-43.0%)reduction for homelessness and 39.0% (21.1-75.2%)for displacement.When both are removed the result is similar, with a 64.9% (48.2-77.9%)reduction in violence.